Direct Coexistence Simulations Research Articles

Phase diagram of NaCl–water by computer simulations: performance of non-polarizable force-fields

This study evaluates the capabilities of two new ion force fields (Tanaka [Yagasaki et al. JCTC 2020] and Madrid-2019 [Zerón et al. JCP 2019]) in conjunction with the TIP4P/2005 model for water in reproducing the NaCl–water phase diagram using molecular dynamics direct coexistence simulations. The performance of those force-fields is critically compared with the phase diagram recently reported for the Joung Cheatham-TIP4P/2005 force-field [Bianco et al. JCP 2022]. Excellent agreement between the three force-fields and experiments is found for the ice-solution line due to the dominance of the water model in the location of this line. For the NaCl-solution line, the Tanaka and the Madrid-2019 models accurately predict solubility at ambient temperature, but they underestimate somewhat the experimental line as temperature increases. The JC model underestimates NaCl solubility both at ambient and at high temperatures. All models underestimate the stability of the NaCl·2H 2 O solid, which remains metastable with respect to NaCl. Equilibration of the NaCl·2H 2 O-solution system is challenging at low temperatures due to the slow diffusion of concentrated NaCl solutions. The phase diagrams obtained serve as a foundation for studying crystal nucleation, and solid-solution interfaces, and provide very valuable information for the development of imporved NaCl–H 2 O force-fields.

A simple and accurate method to determine fluid-crystal phase boundaries from direct coexistence simulations.

One method for computationally determining phase boundaries is to explicitly simulate a direct coexistence between the two phases of interest. Although this approach works very well for fluid-fluid coexistences, it is often considered to be less useful for fluid-crystal transitions, as additional care must be taken to prevent the simulation boundaries from imposing unwanted strains on the crystal phase. Here, we present a simple adaptation to the direct coexistence method that nonetheless allows us to obtain highly accurate predictions of fluid-crystal coexistence conditions, assuming that a fluid-crystal interface can be readily simulated. We test our approach on hard spheres, the screened Coulomb potential, and a 2D patchy-particle model. In all cases, we find excellent agreement between the direct coexistence approach and (much more cumbersome) free-energy calculation methods. Moreover, the method is sufficiently accurate to resolve the (tiny) free-energy difference between the face-centered cubic and hexagonally close-packed crystal of hard spheres in the thermodynamic limit. The simplicity of this method also ensures that it can be trivially implemented in essentially any simulation method or package. Hence, this approach provides an excellent alternative to free-energy based methods for the precise determination of phase boundaries.

Three-phase equilibria of hydrates from computer simulation. II. Finite-size effects in the carbon dioxide hydrate.

In this work, the effects of finite size on the determination of the three-phase coexistence temperature (T3) of the carbon dioxide (CO2) hydrate have been studied by molecular dynamic simulations and using the direct coexistence technique. According to this technique, the three phases involved (hydrate-aqueous solution-liquid CO2) are placed together in the same simulation box. By varying the number of molecules of each phase, it is possible to analyze the effect of simulation size and stoichiometry on the T3 determination. In this work, we have determined the T3 value at 8 different pressures (from 100 to 6000bar) and using 6 different simulation boxes with different numbers of molecules and sizes. In two of these configurations, the ratio of the number of water and CO2 molecules in the aqueous solution and the liquid CO2 phase is the same as in the hydrate (stoichiometric configuration). In both stoichiometric configurations, the formation of a liquid drop of CO2 in the aqueous phase is observed. This drop, which has a cylindrical geometry, increases the amount of CO2 available in the aqueous solution and can in some cases lead to the crystallization of the hydrate at temperatures above T3, overestimating the T3 value obtained from direct coexistence simulations. The simulation results obtained for the CO2 hydrate confirm the sensitivity of T3 depending on the size and composition of the system, explaining the discrepancies observed in the original work by Míguez et al. [J. Chem Phys. 142, 124505 (2015)]. Non-stoichiometric configurations with larger unit cells show a convergence of T3 values, suggesting that finite-size effects for these system sizes, regardless of drop formation, can be safely neglected. The results obtained in this work highlight that the choice of a correct initial configuration is essential to accurately estimate the three-phase coexistence temperature of hydrates by direct coexistence simulations.

Three-phase equilibria of hydrates from computer simulation. I. Finite-size effects in the methane hydrate.

Clathrate hydrates are vital in energy research and environmental applications. Understanding their stability is crucial for harnessing their potential. In this work, we employ direct coexistence simulations to study finite-size effects in the determination of the three-phase equilibrium temperature (T3) for methane hydrates. Two popular water models, TIP4P/Ice and TIP4P/2005, are employed, exploring various system sizes by varying the number of molecules in the hydrate, liquid, and gas phases. The results reveal that finite-size effects play a crucial role in determining T3. The study includes nine configurations with varying system sizes, demonstrating that smaller systems, particularly those leading to stoichiometric conditions and bubble formation, may yield inaccurate T3 values. The emergence of methane bubbles within the liquid phase, observed in smaller configurations, significantly influences the behavior of the system and can lead to erroneous temperature estimations. Our findings reveal finite-size effects on the calculation of T3 by direct coexistence simulations and clarify the system size convergence for both models, shedding light on discrepancies found in the literature. The results contribute to a deeper understanding of the phase equilibrium of gas hydrates and offer valuable information for future research in this field.

A streamlined molecular-dynamics workflow for computing solubilities of molecular and ionic crystals.

Computing the solubility of crystals in a solvent using atomistic simulations is notoriously challenging due to the complexities and convergence issues associated with free-energy methods, as well as the slow equilibration in direct-coexistence simulations. This paper introduces a molecular-dynamics workflow that simplifies and robustly computes the solubility of molecular or ionic crystals. This method is considerably more straightforward than the state-of-the-art, as we have streamlined and optimised each step of the process. Specifically, we calculate the chemical potential of the crystal using the gas-phase molecule as a reference state, and employ the S0 method to determine the concentration dependence of the chemical potential of the solute. We use this workflow to predict the solubilities of sodium chloride in water, urea polymorphs in water, and paracetamol polymorphs in both water and ethanol. Our findings indicate that the predicted solubility is sensitive to the chosen potential energy surface. Furthermore, we note that the harmonic approximation often fails for both molecular crystals and gas molecules at or above room temperature, and that the assumption of an ideal solution becomes less valid for highly soluble substances.

Prediction of aqueous solubility of a strongly soluble solute from molecular simulation.

The prediction of solubilities of compounds by means of molecular simulation has been receiving increasing attention due to the key role played by solubility in countless applications. We have predicted the aqueous solubility of urea at 300K from chemical potential calculations for two urea model combinations: Özpinar/TIP3P and Hölzl/(TIP4P/2005). The methodology assumes that the intramolecular contribution of the urea molecule to the chemical potentials is identical in the crystal and in solution and, hence, cancels out. In parallel to the chemical potential calculations, we also performed direct coexistence simulations of a urea crystal slab in contact with urea-water solutions with the aim to identify upper and lower bounds to the solubility value using an independent route. The chemical potential approach yielded similar solubilities for both urea models, despite the actual chemical potential values showing a significant dependence on the force field. The predicted solubilities for the two models were 0.013-0.018 (Özpınar) and 0.008-0.012 (Hölzl) mole fraction, which are an order of magnitude lower than the experimental solubility that lies in a range of 0.125-0.216 mole fraction. The direct coexistence solubility bounds were relatively wide and did not encompass the chemical potential based solubilities, although the latter were close to the lower bound values.

Phase diagrams-Why they matter and how to predict them.

Understanding the thermodynamic stability and metastability of materials can help us to, for example, gauge whether crystalline polymorphs in pharmaceutical formulations are likely to be durable. It can also help us to design experimental routes to novel phases with potentially interesting properties. In this Perspective, we provide an overview of how thermodynamic phase behavior can be quantified both in computer simulations and machine-learning approaches to determine phase diagrams, as well as combinations of the two. We review the basic workflow of free-energy computations for condensed phases, including some practical implementation advice, ranging from the Frenkel-Ladd approach to thermodynamic integration and to direct-coexistence simulations. We illustrate the applications of such methods on a range of systems from materials chemistry to biological phase separation. Finally, we outline some challenges, questions, and practical applications of phase-diagram determination which we believe are likely to be possible to address in the near future using such state-of-the-art free-energy calculations, which may provide fundamental insight into separation processes using multicomponent solvents.

Phase diagram of the TIP4P/Ice water model by enhanced sampling simulations.

We studied the phase diagram for the TIP4P/Ice water model using enhanced sampling molecular dynamics simulations. Our approach is based on the calculation of ice-liquid free energy differences from biased coexistence simulations that reversibly sample the melting and growth of layers of ice. We computed a total of 19 melting points for five different ice polymorphs, which are in excellent agreement with the melting lines obtained from the integration of the Clausius-Clapeyron equation. For proton-ordered and fully proton-disordered ice phases, the results are in very good agreement with previous calculations based on thermodynamic integration. For the partially proton-disordered ice III, we find a large increase in stability that is in line with previous observations using direct coexistence simulations for the TIP4P/2005 model. This issue highlights the robustness of the approach employed here for ice polymorphs with diverse degrees of proton disorder. Our approach is general and can be applied to the calculation of other complex phase diagrams.

Machine-learning effective many-body potentials for anisotropic particles using orientation-dependent symmetry functions.

Spherically symmetric atom-centered descriptors of atomic environments have been widely used for constructing potential or free energy surfaces of atomistic and colloidal systems and to characterize local structures using machine learning techniques. However, when particle shapes are non-spherical, as in the case of rods and ellipsoids, standard spherically symmetric structure functions alone produce imprecise descriptions of local environments. In order to account for the effects of orientation, we introduce two- and three-body orientation-dependent particle-centered descriptors for systems composed of rod-like particles. To demonstrate the suitability of the proposed functions, we use an efficient feature selection scheme and simple linear regression to construct coarse-grained many-body interaction potentials for computationally efficient simulations of model systems consisting of colloidal particles with an anisotropic shape: mixtures of colloidal rods and non-adsorbing polymer coils, hard rods enclosed by an elastic microgel shell, and ligand-stabilized nanorods. We validate the machine-learning (ML) effective many-body potentials based on orientation-dependent symmetry functions by using them in direct coexistence simulations to map out the phase behavior of colloidal rods and non-adsorbing polymer coils. We find good agreement with the results obtained from simulations of the true binary mixture, demonstrating that the effective interactions are well described by the orientation-dependent ML potentials.

Machine learning many-body potentials for colloidal systems.

Simulations of colloidal suspensions consisting of mesoscopic particles and smaller species such as ions or depletants are computationally challenging as different length and time scales are involved. Here, we introduce a machine learning (ML) approach in which the degrees of freedom of the microscopic species are integrated out and the mesoscopic particles interact with effective many-body potentials, which we fit as a function of all colloid coordinates with a set of symmetry functions. We apply this approach to a colloid-polymer mixture. Remarkably, the ML potentials can be assumed to be effectively state-independent and can be used in direct-coexistence simulations. We show that our ML method reduces the computational cost by several orders of magnitude compared to a numerical evaluation and accurately describes the phase behavior and structure, even for state points where the effective potential is largely determined by many-body contributions.

Computing the Liquidus of Binary Monatomic Salt Mixtures with Direct Simulation and Alchemical Free Energy Methods.

We describe and validate a free-energy-based method for computing the liquidus for binary solid-liquid phase diagrams in molecular simulations of monatomic salts. The method is demonstrated by calculating the liquidus for LiCl-KCl and MgCl2-KCl salt mixtures with the polarizable ion model (PIM). The free-energy-based method is cross-validated with direct coexistence simulations. Both techniques show excellent agreement with one another. Though the predictions of the PIM disagree with experiments, we use our free-energy-based approach to decouple the contributions of liquid mixture nonidealities and pure component solid-liquid equilibrium to the phase diagram. In both mixtures, the PIM accurately reproduces the liquid phase nonidealities but fails to predict the liquidus because it does not accurately predict the pure component melting temperature of LiCl or MgCl2.

Computation of the chemical potential and solubility of amorphous solids.

Using a recently developed technique to estimate the equilibrium free energy of glassy materials, we explore if equilibrium simulation methods can be used to estimate the solubility of amorphous solids. As an illustration, we compute the chemical potentials of the constituent particles of a two-component Kob-Andersen model glass former. To compute the chemical potential for different components, we combine the calculation of the overall free energy of the glass with a calculation of the chemical potential difference of the two components. We find that the standard method to compute chemical potential differences by thermodynamic integration yields not only a wide scatter in the chemical potential values, but also, more seriously, the average of the thermodynamic integration results is well above the extrapolated value for the supercooled liquid. However, we find that if we compute the difference in the chemical potential of the components with the non-equilibrium free-energy expression proposed by Jarzynski, we obtain a good match with the extrapolated value of the supercooled liquid. The extension of the Jarzynski method that we propose opens a potentially powerful route to compute the free-energy related equilibrium properties of glasses. We find that the solubility estimate of amorphous materials obtained from direct-coexistence simulations is only in fair agreement with the solubility prediction based on the chemical potential calculations of a hypothetical "well-equilibrated glass." In direct-coexistence simulations, we find that, in qualitative agreement with experiments, the amorphous solubility decreases with time and attains a low solubility value.

Spontaneous NaCl-doped ices Ih, Ic, III, V and VI. Understanding the mechanism of ion inclusion and its dependence on the crystalline structure of ice.

Direct coexistence simulations on a microsecond time scale have been performed for different types of ice (Ih, Ic, III, V, and VI) in contact with a NaCl aqueous solution at different pressures. In line with the previous results obtained for ice Ih [Conde et al., Phys. Chem. Chem. Phys., 2017, 19, 9566-9574], our results reveal the spontaneous growth of a new ice doped phase and the formation of a brine rejection phase in all ices studied. However, both the preferential incorporation of ions into the ice lattice and the inclusion mechanisms depend on the crystalline structure of each ice. This work shows the inclusion of Cl- and Na+ ions in ice from salt using molecular dynamics simulation, in agreement with the experimental evidence found in the literature. The model used for water is TIP4P/2005. For NaCl we employ a set of potential parameters that uses unit charges for the ions.

Scaled charges at work: Salting out and interfacial tension of methane with electrolyte solutions from computer simulations

The solubility of methane in water decreases when a small amount of salt is present. This is usually denoted as the salting out effect (i.e., the methane is expelled from the solution when it contains small amounts of salt). The effect is important, for instance the solubility is reduced by a factor of three in a 4 m (mol/kg) NaCl solution. Some years ago we showed that the salting out effect of methane in water can be described qualitatively by molecular models using computer simulations. However the salting out effect was overestimated. In fact, it was found that the solubility of methane was reduced by a factor of eight. This points to limitations in the force field used. In this work we have carried out direct coexistence simulations to describe the salting out effect of methane in water using a recently proposed force field (denoted as Madrid-2019) based on the use of scaled charges for the ions and the TIP4P/2005 force field for water. For NaCl the results of the Madrid-2019 force field significantly improve the description of salting out of methane. For other salts the results are quite reasonable. Thus the reduction of the charge of the ions also seems to be able to improve the description of salting out effect of methane in water. Besides this we shall show that the brine-methane interface exhibits an increased interfacial tension as compared to that of the water-methane system. It is well known that electrolytes tend to increase the surface tension of liquid water, and this seems also to be the case for the interface between water and methane.

Solubilities of pyrene in organic solvents: Comparison between chemical potential calculations using a cavity-based method and direct coexistence simulations

In this paper, we benchmark a cavity-based simulation method for calculating the relative solubility of large molecules in explicit solvents. The essence of the procedure is the accounting of the Gibbs energy change associated with an alchemical thermodynamic cycle where, in sequence, a cavity is created in a solvent, a solute is inserted in the cavity and the cavity is annihilated. The Gibbs energy change is equated to the excess chemical potential allowing the comparison of solubilities in different solvents. The results obtained using the cavity-based method are compared to direct large-scale molecular dynamics simulations performed using coarse-grained models for calculating the partition coefficient of pyrene between heptane and toluene. We demonstrate the applicability of this cavity-based technique under high pressure/temperature conditions.

A simulation study of homogeneous ice nucleation in supercooled salty water

We use computer simulations to investigate the effect of salt on homogeneous ice nucleation. The melting point of the employed solution model was obtained both by direct coexistence simulations and by thermodynamic integration from previous calculations of the water chemical potential. Using a seeding approach, in which we simulate ice seeds embedded in a supercooled aqueous solution, we compute the nucleation rate as a function of temperature for a 1.85 NaCl mol per water kilogram solution at 1 bar. To improve the accuracy and reliability of our calculations, we combine seeding with the direct computation of the ice-solution interfacial free energy at coexistence using the Mold Integration method. We compare the results with previous simulation work on pure water to understand the effect caused by the solute. The model captures the experimental trend that the nucleation rate at a given supercooling decreases when adding salt. Despite the fact that the thermodynamic driving force for ice nucleation is higher for salty water for a given supercooling, the nucleation rate slows down with salt due to a significant increase of the ice-fluid interfacial free energy. The salty water model predicts an ice nucleation rate that is in good agreement with experimental measurements, bringing confidence in the predictive ability of the model. We expect that the combination of state-of-the-art simulation methods here employed to study ice nucleation from solution will be of much use in forthcoming numerical investigations of crystallization in mixtures.

On the calculation of solubilities via direct coexistence simulations: Investigation of NaCl aqueous solutions and Lennard-Jones binary mixtures.

Direct coexistence molecular dynamics simulations of NaCl solutions and Lennard-Jones binary mixtures were performed to explore the origin of reported discrepancies between solubilities obtained by direct interfacial simulations and values obtained from the chemical potentials of the crystal and solution phases. We find that the key cause of these discrepancies is the use of crystal slabs of insufficient width to eliminate finite-size effects. We observe that for NaCl crystal slabs thicker than 4 nm (in the direction perpendicular to the interface), the same solubility values are obtained from the direct coexistence and chemical potential routes, namely, 3.7 ± 0.2 molal at T = 298.15 K and p = 1 bar for the JC-SPC/E model. Such finite-size effects are absent in the Lennard-Jones system and are likely caused by surface dipoles present in the salt crystals. We confirmed that μs-long molecular dynamics runs are required to obtain reliable solubility values from direct coexistence calculations, provided that the initial solution conditions are near the equilibrium solubility values; even longer runs are needed for equilibration of significantly different concentrations. We do not observe any effects of the exposed crystal face on the solubility values or equilibration times. For both the NaCl and Lennard-Jones systems, the use of a spherical crystallite embedded in the solution leads to significantly higher apparent solubility values relative to the flat-interface direct coexistence calculations and the chemical potential values. Our results have broad implications for the determination of solubilities of molecular models of ionic systems.

A unified approach to computation of solid and liquid free energy to revisit the solid-fluid equilibrium of Lennard-Jones chains.

Liquid free energies are computed by integration along a path from a reference system of known free energy, using a strong localization potential. A particular choice of localization pathway is introduced, convenient for use in molecular dynamics codes, and which achieves accurate results without the need to include the identity-swap or relocation Monte Carlo moves used in previous studies. Moreover, an adaptive timestep is introduced to attain the reference system. Furthermore, a center-of-mass correction that is different from previous studies and phase-independent is incorporated. The resulting scheme allows computation of both solid and liquid free energies with only minor differences in simulation protocol. This is used to re-visit solid-liquid equilibrium in a system of short semi-flexible Lennard-Jones chain molecules. The computed melting curve is demonstrated to be consistent with direct co-existence simulations and computed hysteresis loops, provided that an entropic term arising from unsampled solid states is included.

The effect of dispersion interactions on the properties of LiF in condensed phases

Classical molecular dynamics simulations are performed on LiF in the framework of the polarizable ion model. The overlap repulsion and polarization terms of the interaction potential are derived on a purely non-empirical, first-principles basis. For the dispersion, three cases are considered: a first one in which the dispersion parameters are set to zero and two others in which they are included, with different parametrizations. Various thermodynamic, structural and dynamic properties are calculated for the solid and liquid phases. The melting temperature is also obtained from direct coexistence simulations of the liquid and solid phases. Dispersion interactions appear to have an important effect on the densities of both phases and on the melting point, although the liquid properties are not affected when simulations are performed in the NVT ensemble at the experimental density.

Determining the phase diagram of water from direct coexistence simulations: The phase diagram of the TIP4P/2005 model revisited

Direct coexistence simulations between the fluid and solid phases are performed for several ices. For ices Ih and VII it has already been shown that the methodology is successful and the melting point is in agreement with that obtained from free energy calculations. In this work the methodology is applied to ices II, III, V, and VI. The lengths of the direct coexistence runs for the high pressure polymorphs are not too long and last less than 20 ns for all ices except for ice II where longer runs (of about 150 ns) are needed. For ices II, V, and VI the results obtained are completely consistent with those obtained from free energy calculations. However, for ice III it is found that the melting point from direct coexistence simulations is higher than that obtained from free energy calculations, the difference being greater than the statistical error. Since ice III presents partial proton orientational disorder, the departure is attributed to differences in the partial proton order in the water model with respect to that found in the experiment. The phase diagram of the TIP4P/2005 model is recalculated using the melting points obtained from direct coexistence simulations. The new phase diagram is similar to the previous one except for the coexistence lines where ice III is involved. The range of stability of ice III on the p-T plot of the phase diagram increases significantly. It is seen that the model qualitatively describes the phase diagram of water. In this work it is shown that the complete phase diagram of water including ices Ih, II, III, V, VI, VII, and the fluid phase can be obtained from direct coexistence simulations without the need of free energy calculations.